Research Question: Do medical consultants lead patients to better outcomes?
How many orange Reeses Pieces are in a typical bag of 40?
https://www.rossmanchance.com/applets/2021/oneprop/OneProp.htm?candy=1
Using the applet, set the Probability of orange to 0.5, the Number of candies to 40, the Number of samples to 1, and choose the Proportion of orange statistic.
Click Draw Samples. How many orange did you get? What is \(\hat{p}_\text{boot}\)? Does it equal \(\hat{p}\)? Is it close?
Now change Number of samples to 99 and click Draw Samples again. When it finishes, make sure Total Samples is 100.
The dotplot that appears should have 100 dots. Use this dotplot to estimate a bootstrap 90% confidence interval for the proportion of orange Reeses Pieces.
In a sample of 120 listeners, only 3 were able to correctly guess the tune that was being tapped. (\(\hat{p} = 0.025\))
Simulation:
\[ 0.0417 \quad 0.025 \quad 0.025 \quad 0.0083 \quad 0.05 \quad 0.0333 \quad 0.025 \quad 0 \quad 0.0083 \quad 0 \]
Reload the applet, set the Probability of orange to 0.5, the Number of candies to 200, and choose the Proportion of orange statistic. You probably also want to uncheck Show animation. Now change Number of samples to 100 and click Draw Samples again. When it finishes, make sure Total Samples is 100.
Use this dotplot to estimate a bootstrap 90% confidence interval for the proportion of orange Reeses Pieces.
Compare your confidence interval from #2 with your confidence interval from the last activity. How does changing the sample size affect the confidence interval?
Now use the same plot of bootstrapped \(\hat{p}_\text{boot}\)’s to estimate an 80% confidence interval. How does changing the confidence level change the confidence interval?
Experiment with some different values for Probability of orange. How does changing the observed \(\hat{p}\) affect the confidence interval?